UNBOUNDED REPRESENTATIONS OF q-DEFORMATION OF

arXiv:0806.3503v1 [math.QA] 21 Jun 2008

UNBOUNDED REPRESENTATIONS OF q-DEFORMATION OF CUNTZ ALGEBRA VASYL OSTROVSKYI, DANIIL PROSKURIN, AND LYUDMILA TUROWSKA To the memory of Leonid Vaksman Abstract. We study a deformation of the Cuntz-Toeplitz C ∗ -algebra determined by the relations a∗i ai = 1 + qai a∗i , a∗i aj = 0. We define well-behaved unbounded ∗-representations of the ∗-algebra defined by relations above and classify all such irreducible representations up to unitary equivalence.

Introduction Many of the structures that have been studied recently arise as deformations of classical objects, e.g. deformations of the canonical commutation relations (CCR) and the canonical anti-commutation relations (CAR), quantum groups, quantum homogeneous spaces, non-commutative probability etc (see [3, 5, 9, 15, 21, 23, 24]). From the physical point of view important classes of objects come from the Fock space formalism forming algebras generated by raising and lowering operators and their numerous generalizations. Examples of such generalizations include qdeformed quantum oscillator algebra ([3, 9, 15]), twisted CCR ([21]), generalised deformed oscillator ([8]) and more general quadratic algebras with Wick ordering ([14]). Here unbounded representations arise naturally since most of the physical observables can not be realized by bounded operators. During the past 30 years many works concerning (topological) algebras of unbounded operators and their physical applications appeared in the literature (see e.g. [1, 10, 12, 22] and references therein). One of the most known q-deformations of CCR is q-CCR with one degree of freedom, see [3, 15], i.e. the ∗-algebra which is generated by elements a, a∗ satisfying the commutation relation a∗ a = 1 + qaa∗ ,

(1)

where q ∈ [0, 1). For many degree of freedom there exist several versions of q-CCR algebras, see [4, 9, 16, 21]. In this paper we consider a subclass of qij -CCR algebras introduced in [4], namely ∗-algebras, denoted later by Onq , which are generated by ai , a∗i , i = 1, . . . , n, subject to the relations (2)

a∗i ai = 1 + qai a∗i ,

a∗i aj = 0,

i 6= j, i, j = 1, . . . , n, 0 < q < 1.

2000 Mathematics Subject Classification. Primary 47L60, 47L30, 47A67 Secondary 81R10. Key words and phrases. Cuntz algebra, deformed commutation relations, unbounded representation. 1

2

VASYL OSTROVSKYI, DANIIL PROSKURIN, AND LYUDMILA TUROWSKA

Note that for q = 0 we obtain the ∗-algebra, On0 , generated by n isometries with orthogonal ranges, i.e. On0 = C hsi , s∗i sj = δij 1, i, j = 1, . . . , ni . Its enveloping C ∗ -algebra is an extension of the Cuntz C ∗ -algebra On by compact operators whose representation theory was extensively studied (see [6, 7] and references therein). Our aim is to describe irreducible unbounded representations of Onq , n > 1. Note that each Onq , n > 1, has also bounded representations, however, since the corresponding universal enveloping C ∗ -algebra Onq is isomorphic to the Cuntz-Toeplitz C ∗ -algebra On0 (see Section 1), Onq is not of type I algebra and the problem of unitary classification of its irreducible ∗-representations is complicated. Nevertheless it turns out to be possible to classify all its “well-behaved” unbounded irreducible ∗-representations up to unitary equivalence. Unbounded representations are known to be a very delicate thing, since unbounded operators are not defined on the whole space. Depending on chosen domains of representations they can behave differently (see, e.g. [18, 22]). In the theory of ∗-representations of finite-dimensional Lie algebras the class of well-behaved representations form the representations which can be integrated to unitary representations of the corresponding simply connected Lie group. Nelson’s fundamental theorem (see [2, 18]) gives a criterion for the integrability in terms of the Laplace operator of the Lie algebra, requiring its essential self-adjointness on a common invariant dense domain. Our definition of well-behaved representation is motivated by this issue. The well-known Stone-von Neumann theorem says that up to unitary equivalence there exists a unique irreducible representation of CCR which is unbounded and given by raising and lowering operators; the representation is often called the Fock representation. However, for q-CCR and, as we will see, for Onq , q ∈ (0, 1), the Fock representation is bounded, and a whole bunch of irreducible unbounded representations which do not have any classical analogs, arises (for q-CCR see for example [19]). The paper is organized as follows. In Preliminaries we recall a classification of irreducible representations of O1q (or q-CCR) and prove an isomorphism of the enveloping C ∗ -algebras of Onq , q ∈ (0, 1), to that of On0 . As a consequence we get that the description of bounded ∗-representations of Onq is equivalent to that of ∗-representations of the Cuntz-Toeplitz algebra On0 . In Section 2.1 we give a definition of well-behaved unbounded ∗-representation of Onq in spirit of [21] and also present an equivalent one in terms of bounded operators. Finally, in Section 2.2 we obtain a classification of all irreducible well-behaved unbounded ∗-representations of Onq up to unitary equivalence. 1. Preliminaries Recall some facts on representation theory and properties of the universal enveloping C ∗ -algebra of q-CCR or O1q , 0 ≤ q < 1, see [20]. If q = 0 we get the well-known ∗-algebra generated by a single isometry. Obviously any representation of O10 is bounded. Moreover, any irreducible representation of O10 , up to a unitary equivalence, is either one-dimensional a1 = exp ıϕ, ϕ ∈ [0, 2π), or the

UNBOUNDED REPRESENTATIONS OF q-DEFORMATION OF CUNTZ ALGEBRA

3

infinite-dimensional, called also the Fock representation, given by the action a1 en = en+1 ,

n ∈ N,

on an orthonormal basis {en : n ∈ N} in l2 (N). When 0 < q < 1, unbounded representations will also arise. Defining “wellbehaved” unbounded representations as in Section 2.1 one has the following Proposition 1. Any irreducible representation of O1q with 0 < q < 1 is unitarily equivalent to exactly one listed below. 1. The Fock representation acting on l2 (N): r 1 − qn en+1 , n ∈ N. πF (a1 ) en = 1−q 2. One-dimensional representations: r 1 πϕ (a1 ) e = exp(iϕ) e, 1−q

ϕ ∈ [0, 2π).

3. Unbounded representations acting on l2 (Z) : r 1 − qn πx (a1 ) en = + q n x en+1 , n ∈ Z, x ∈ (1 + qx0 , x0 ] 1−q where x0 >

1 1−q

is fixed.

One of the fundamental facts on representation theory of O1 is the Wold decomposition theorem, stating that any isometric operator is an orthogonal direct sum of a multiple of the unilateral shift and a unitary operator (see [17]). Using the description of irreducible representations of O1q one can get a generalization of the Wold decomposition theorem to the case of linear operator satisfying q-canonical commutation relation (below we will refer to this fact as the q-Wold decomposition theorem). Theorem 1. Let A : H → H be a bounded linear operator satisfying for some q ∈ (0, 1) the q-commutation relation A∗ A = 1 + qAA∗ . Then H can be decomposed into orthogonal sum of subspaces H = H0 ⊕Hu invariant is a multiple with respect to the actions of A, A∗ and such that the restriction A|H0 q of the weighted shift on l2 (Z+ ) defined on the standard basis by Aen =

and A|Hu =

√ 1 U, 1−q

1−qn 1−q en+1 ,

where U is a unitary operator on Hu .

If we do not assume an operator A satisfying (1) to be bounded we can still decompose H into orthogonal direct sum of invariant subspaces H0 and Hu such that the restriction of A to H0 is a multiple of the Fock representation. However in general situation Hu is decomposed into orthogonal sum Hu = H1 ⊕ H2 , where 1 U with unitary U , and the restriction A|H2 is unbounded and given A|H1 = √1−q by a direct integral of unbounded irreducible representations. In particular, in the polar decomposition A|H2 = SC the isometric part S is unitary. Note that the Fock representation πF of O1q is faithful and the universal enveloping C ∗ -algebra of O1q is isomorphic to the C ∗ -algebra generated by πF (a1 ).

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Definition 1. Let A be a ∗-algebra. Assume that the set, Rep A, of all its bounded representations is not empty and kπ(a)k < ∞

sup π∈Rep A

for any a ∈ A. The universal enveloping C ∗ -algebra of a ∗-algebra A is the completion of A/R with respect to the following norm ka + Rk =

sup

kπ(a)k,

π∈Rep A

where R = {a ∈ A | π(a) = 0, π ∈ Rep A}. The existence of the enveloping C ∗ -algebra of Onq follows from the fact that 1 kπ(ai )k2 ≤ 1−q , i = 1, . . . , n for any bounded representation π of Onq (see Theorem 1). In what follows we denote this C ∗ -algebra by Onq . It is known that O1q ≃ O10 for any q ∈ (−1, 1), see for example [11]. The same is true for Onq , n ∈ N. Theorem 2. Onq ≃ On0 for any q ∈ (−1, 1). Proof. Suppose that Onq is realized by linear operators on a Hilbert space. Consider the polar decompositions ai = si ci , i = 1, . . . , d, where c2i = a∗i ai and si are partial isometries such that ker si = ker ci . Since the spectrum σ(c2i ) of c2i is n 1 { 1−q 1−q , n ∈ N} ∪ { 1−q }, each ci is invertible and therefore each si is an isometry. 1

Moreover, si = ai (a∗i ai )− 2 ∈ C ∗ (ai , a∗i ), i = 1, . . . , n. Further from a∗i aj = 0 one has ci s∗i sj cj = 0, hence s∗i sj = 0, i 6= j. Since in any irreducible bounded representation of O1q with −1 < q < 1 one has ∞ X a1 = s1 , q n sn1 s∗n 1 n=0

the same equality holds in Onq for ai and si , i = 1, . . . , n. Therefore ai ∈ C ∗ (si , s∗i , i = 1, . . . , n), i = 1, . . . , n, giving the statement of the theorem. Since the Cuntz-Toeplitz algebra On0 is a not of type I algebra (see [6]), this theorem shows that the classification problem of all irreducible representations of Onq and therefore all irreducible bounded representations of Onq is very complicated. 2. Unbounded representations of Onq 2.1. Definition and properties. We will start with defining well-behaved representations of Onq . Definition 2. We say that a family {Ai , i = 1, . . . , n} of closed operators on a Hilbert space H defines a well-behaved representation π of Onq if (1) there exists a dense linear subset D ⊂ H which is invariant with respect to A1 , . . . , An , A∗1 , . . . , A∗n ; (2) A∗i Ai x = (1 + qAi A∗i )x, A∗i Aj x = i 6= j, i = 1, . . . , n, x ∈ D; P0, n (3) the positive linear operator ∆ = i=1 A∗i Ai is essentially selfadjoint on D.


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This definition is similar to the definition of unbounded representations of twisted canonical commutation relations given by Pusz and Woronowicz ([21]). It is motivated by the Nelson criterion of the integrability for representations of Lie algebras ([2, 18]). Next two theorems provide a criteria for representations to be well-behaved in terms of bounded operators. It will be important for later classification of wellbehaved (irreducible) representations of Onq . Before stating the theorems we recall the definition of analytic and bounded vectors for an unbounded operators, which will be used in the proofs. If A is an operator in a Hilbert space H, then u ∈ H is said to be an analytic vector (a bounded vector) for A, if ∞ X ||An u|| n s < ∞, n! n=0

for some s > 0 (||An u|| ≤ C n , for some C > 0 respectively). These concepts are fundamental in the theory of integrable representations of Lie algebras ([2, 13, 18]). Theorem 3. Let {Ai , i = 1, . . . , n} be a family of linear operators on H defining a well-behaved representation of Onq and let Ai = Si |Ai | be the polar decomposition of Ai and Di = Si |Ai |Si∗ , i = 1, . . . , n. Then (a) f (Di2 )Si = Si f (1 + qDi2 ) for any real bounded measurable function f ; (b) for any i, j the operators |Ai |, |Aj | commute in the sense of resolutions of identity; (c) Si∗ Sj = δij I. Proof. Let Cj2 be the closure of A∗j Aj on D, j = 1, . . . , n. Clearly, Cj2 is symmetric. In order to show that all Cj2 are selfadjoint and mutually commute in the sense of resolution of identity we prove first that ∆n Cj2 y = Cj2 ∆n y, y ∈ D(∆n+2 ).

(3)

Here and subsequently, D(a) denotes the domain of an operator a. We have Ci2 Cj2

= (1 + qAi A∗i )(1 + qAj A∗j ) = (1 + qAi A∗i + qAj A∗j ) = (1 + qA∗j Aj )(1 + qAi A∗i ) = Cj2 Ci2

on D. Thus if x, y ∈ D then (∆x, Cj2 y) = (Cj2 x, ∆y) and (Cj2 ∆x, y) = (Cj2 x, ∆y).

(4)

As D is a core for ∆ the second equality holds also for any y ∈ D(∆). We shall show next that D(Cj2 ) ⊃ D(∆2 ). In fact, by (4) n n n X X X X Ci4 + (1 + qAi A∗i + qAj A∗j ) Ci2 ) = Ci2 )( ∆2 = ( i=1

2

i=1

i=1

i6=j

Cj4

giving ∆ ≥ on D. Since D is a core for ∆2 we have that for y ∈ D(∆2 ) there exists {yn } ∈ D such that yn → y, ∆2 yn → ∆2 y and ||Cj2 (yn − ym )|| =

(Cj2 (yn − ym ), Cj2 (yn − ym ))

=

(Cj4 (yn − ym ), yn − ym )

≤

(∆2 (yn − ym ), yn − ym ).

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Thus the sequence {Cj2 yn } converges to some z ∈ H and by the closededness of Cj2 , y ∈ D(Cj2 ) and z = Cj2 y. Moreover, Cj4 ≤ ∆2 on D(∆2 ).

(5)

Let now y ∈ D(∆3 ). Then ∆y ∈ D(∆2 ) ⊂ D(Cj2 ), y ∈ D(∆2 ) ⊂ D(Cj2 ) and by (4) (∆x, Cj2 y) = (x, Cj2 ∆y), x ∈ D. By the closededness argument the same equality holds for any x ∈ D(∆) giving Cj2 D(∆3 ) ⊂ D(∆) and ∆Cj2 = Cj2 ∆ on D(∆3 ). We proceed now by induction and suppose that for any y ∈ D(∆n ), n ≥ 3, Cj2 y ∈ D(∆n−2 ) and ∆n−2 Cj2 y = Cj2 ∆n−2 y.

(6)

In particular, if y ∈ D(∆n+1 ) then Cj2 y ∈ D(∆n−2 ) and (6) holds. Let z = ∆n−2 y. Then z ∈ D(∆3 ) and Cj2 z = ∆n−2 Cj2 y ∈ D(∆). Therefore Cj2 y ∈ D(∆n−1 ) and ∆n−1 Cj2 y = ∆∆n−2 Cj2 y = ∆Cj2 ∆n−2 y = Cj2 ∆n−1 y. Then by induction we obtain (3) for any n ≥ 1. Let Dω be the space of analytic vectors for ∆. As Dω ⊂ D(∆k ), for any k ∈ N, by (5) and (6) ||∆n Cj2 x||2

=

||Cj2 ∆n x||2 = (Cj2 ∆n x, Cj2 ∆n x) = (∆n Cj4 ∆n x, x)

≤

(∆n ∆2 ∆n x, x) = ||∆n+1 x||2 .

This shows that Cj2 Dω ⊂ Dω and, moreover, Cj2 mutually commute on Dω . The last can be seen by computing the scalar product of Ci2 Cj2 x − Cj2 Ci2 x, x ∈ D, with y ∈ Dω . Next we show that any x ∈ Dω is also analytic for all Ci2 . In fact, as Ci4 ≤ ∆2 on Dω , by assuming by induction that Ci4n ≤ ∆2n on Dω we obtain 4(n+1)

Ci

=

Ci2 Ci4n Ci2 ≤ Ci2 ∆4n Ci2

=

∆2n Ci4 ∆2n ≤ ∆2n ∆2 ∆2n = ∆2(n+1) .

This gives ||(Ci2 )n x||2 = (x, Ci4n x) ≤ (x, ∆2n x) ≤ ||∆n x||2 , i.e. Dω is a subset of analytic vectors for all Ci2 . As Dw is invariant with respect to all Ci2 and Ci2 mutually commute on Dω , we have that all Ci2 are selfadjoint and mutually strongly commute, i.e. in the sense of resolutions of identity. In particular, we have proved that each Ci2 is essentially selfadjoint on D, and Ci2 = A∗i Ai = |Ai |2 . Next we prove that f (Ci2 )Si = Si f (1+qCi2 ) for any bounded measurable function f . Let Ri = Ci2 for notation simplicity. As Ri Ai = Ai (1 + qRi ) and Ri A∗i = A∗i (Ri − 1)/q on D, using arguments similar to one given above one can show that for any non-negative integer n and x ∈ D(Rik−1 ) we have that A∗i x ∈ D(Rik ), Ai x ∈ D(Rik ) and Rik Ai x = Ai (1 + qRi )k x and Rik A∗i x = A∗i ((Ri − 1)/q)k x.


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Taking now Di,ω the space of analytic vectors for Ri and using that Ri ≤ (1+Ri )2 we obtain ||Rik Ai x||2

= ||Ai (1 + qRi )k x||2 = ((1 + qRi )k A∗i Ai (1 + qRi )k x, x) ≤ ((1 + qRi )k (1 + Ri )2 (1 + qRi )k x, x) = ((1 + Ri )(1 + qRi )2k (1 + Ri )x, x) ≤ ((1 + Ri )2(k+1) x, x) = ||(1 + Ri )k x||2

giving that Ai Di,ω ⊂ Di,ω . Analogously, one proves that A∗i Di,ω ⊂ Di,ω . Moreover, the relations A∗i Ai = 1 + qAi A∗i and Ri Ai = Ai (1 + qRi ) hold on Di,ω which can be shown analogously to commutation of Ci2 on Dω above. That Di,ω is a core for Ai and A∗i can be proved using the arguments in [21, Proposition 3.3]. The condition f (Ci2 )Si = Si f (1 + qCi2 ) now follows from [19, Theorem 1, Theorem 2]. Furthermore, A∗i Ai = 1 + qAi A∗i ⇔ Ci2 = 1 + qSi Ci2 Si∗ ⇒ Si Si∗ Ci2 = Si Si∗ + qSi Ci2 Si∗ = Ci2 − 1

(7) on D giving

(1 − Si Si∗ )Ci2 = (1 − Si Si∗ ) As D is a core for Ci2 we obtain the equality on D(Ci2 ). Similarly, (1 − Si Si∗ )Ci2n = (1 − Si Si∗ ) on D(Ci2n ) and in particular on Di,ω for any n ≥ 1. The arguments similar to one in [19, Theorem 1] give (1 − Si Si∗ )f (Ci2 ) = f (1)(1 − Si Si∗ ) for any bounded Borel function f . This will also give that Si Si∗ commute with resolution of the identity of Ci and from (7) we will get that Ci ≥ 1 and since ker Ci = ker Si , Si is an isometry. To obtain f (Di2 )Si = Si f (1 + qDi2 ) we note that Ai A∗i = Di2 and (Ai A∗i )Ai = Ai (1 + qAi A∗i ) on Diω . Moreover, clearly, any vector in Di,ω is analytic for Ai A∗i . Using again [19, Theorem 1, Theorem 2] we obtain the desired equality. What is left to prove is that Si∗ Sj = 0 if i 6= j. We have A∗i Aj = Ci Si∗ Sj Cj = 0 on D and (Si∗ Sj Cj x, Ci y) = 0 for any x ∈ D, y ∈ D(Ci ). As each Si is an isometry, the range of Ci is dense implying Si∗ Sj = 0. Remark 1. Let Ej (·) be the resolution of identity of Dj2 = Sj |Aj |2 Sj∗ , j = 1, . . . , n. Then, by [19], (a) is equivalent to (8)

Ej (δ)Sj = Sj Ej (q −1 (δ − 1)), j = 1, . . . , n for any Borel δ ⊂ R.

We will use the following notation when proving our next theorem. Let Λ = {∅, α = (α1 , α2 , . . . , αk ), 1 ≤ αi ≤ n k ∈ N} be the set of all finite multi-indices. Introduce the transformations σ, σk : Λ → Λ, k = 1, . . . , n σ(α1 , . . . , αs ) = σk (α1 , . . . , αs ) =

(α2 , . . . , αs ), σ(α1 ) = ∅, (k, α1 , . . . , αs ).

Below having any family u1 , . . . , un of elements of some algebra and any nonempty multi-index α = (α1 , . . . , αk ) ∈ Λ we will denote by uα the product uα1 uα2 · · · uαk , it will be also convenient for us to put u∅ = 1 Theorem 4. Let Ai = Si |A|i , i = 1, . . . , n, be the polar decompositions of closed operators Ai . If |Ai |, Di , Si , i = 1, . . . , n satisfy conditions (a)-(c) of Theorem 3, then {Ai , i = 1, . . . , n} defines a well-behaved representation of Onq .

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Proof. We construct the necessary domain. The condition (8) implies that given a fixed j, the operators Aj , A∗j form a (well-behaved) representation of q-CCR relation with one degree of freedom. From the generalized Wold decomposition for representations of one-dimensional q-CCR (j) (j) (j) (j) we can write H = H0 ⊕ Hs ⊕ Hu so that in H0 we have a multiple of the (j) (j) Fock representation, in Hs we have Dj2 = (1 − q)−1 and in Hu is unbounded (j)

(j)

component. Notice that in Hs and in Hu the operator Sj is unitary. Let Hj be (j) a span of the vectors Sα x, x ∈ Hu , α ∈ Λ. Lemma 1. 1. Hj , j = 1, . . . , n are invariant subspaces. 2. Hj ⊥ Hk , j 6= k. 3. In the subspace H0 = H ⊖ (H1 ⊕ · · · ⊕ Hn ) the representation is bounded. Proof. 1. It is sufficient to show that each Hj is invariant with respect to Sk , Sk∗ , and Ek (δ), k = 1, . . . , n for any measurable δ. From Sk Sα = Sσk (α) obviously follows the invariance with respect to Sk , k = 1, . . . , n. For α 6= ∅ the invariance with respect to Sk∗ follows as well since Sk∗ Sα = (j) (j) δkα1 Sσ(α) . For the vectors of the form x ∈ Hu , since Sj is unitary in Hu , we (j) have x = Sj Sj∗ x and therefore, Sk∗ x = δjk Sj∗ x ∈ Hu . Take a measurable δ. Since Sk Sk∗ is the projection on the cokernel of Dk2 , then for δ not containing {0} we have Ek (δ)Sj = Ek (δ)Sk Sk∗ Sj = 0, and Ek ({0})sj = (1 − Sk Sk∗ )Sj = Sj , j 6= k. Therefore, for 0 ∈ δ we have Ek (δ)Sj = Sj , and for 0∈ / δ we have Ek (δ)Sj = 0, j 6= k. Thus we have that Ek (δ)Sα x ∈ Hj for α1 6= k. If α1 = k, then we apply (8) and get Ek (δ)Sα x = Sk Ek (q −1 (δ − 1))Sσ(α) x ∈ Hj by induction. It remains to consider the case α = ∅. Again, we can write x = Sj Sj∗ x, and apply the arguments above. (j) (k) 2. Take arbitrary x ∈ Hu , y ∈ Hu , j 6= k. i) Since x = Sj Sj∗ x, y = Sk Sk∗ y, we have (x, y) = (Sj∗ x, Sj∗ Sk Sk∗ y) = 0. ii) For any α = (α1 , . . . , αl ) we have (Sα x, y) = (x, Sα∗ y) = (Sj∗ x, Sj∗ Sα∗ y). But since y = Skl+1 (S ∗ )l+1 y and Sj∗ Sα sl+1 = 0, the latter scalar product is zero. k iii) For any α, β ∈ Λ we have (Sα x, Sβ y) = 0 using quite similar arguments. 3) Is obvious, since unbounded component of any Aj in its Wold decomposition generates Hj . Let us continue the proof of the theorem. By Lemma 1 we decompose the representation space, H = H0 ⊕ H1 · · · ⊕ Hn , in each component we construct the necessary domain Dj , j = 0, . . . , n, and take D = D0 ⊕ D1 ⊕ · · · ⊕ Dn . Since in H0 the operators are bounded, we can take D0 = H0 . We fix some j = 1, . . . , n and construct the corresponding set Dj ⊂ Hj . (j) Let Dj be a span of vectors of the form Sα Ej (δ)x, where x ∈ Hu , α ∈ Λ, δ ⊂ R+ are bounded measurable sets. (j) 1. Dj is dense in Hj . Indeed, choose δ = [0, t], then for any x ∈ Hu , Dj contains xt = Ej ([0, t])x, which converges to x strongly as t → ∞. Applying the operators Sα , α ∈ Λ, we obtain total in Hj set. 2. Dj ⊂ D(Dk2 ), k = 1, . . . , n, and consists of bounded vectors for these operators.


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First we show that for any z ∈ Dj , the sequence Ek ([0, t])Dk2 z converges in Hj as t → ∞. But as noticed above Ek ([0, t])Sl = Ek ({0})Sl , k 6= l, and hence Ek ([0, t])Dk2 Sl = 0, k 6= l. Therefore, Ek ([0, t])Dk2 z = 0 on any z = Sα Ej (δ)x, where α = (α1 , . . . , αm ) with α1 6= k. For α1 = k we have Ek ([0, t])Dk2 z = Sk (1 + qDk2 )Ek ([0, q −1 (t − 1)])Sσ(α) Ej (δ)x and for α with α1 = · · · = αm = k Ek ([0, t])Dk2 Skm Sσm (α) Ej (δ)x (9)

= Skm (

(1 − q)t − 1 + q m 1 − qm ])Sσm (α) Ej (δ)x. I + q m Dk2 )Ek ([0, 1−q (1 − q)q m

If σ m (α) 6= ∅ and αm+1 6= k the expression in (9) is equal to 1 − qm 1 − qm m Sk Sσm (α) Ej (δ)x = Sα Ej (δ)x. 1−q 1−q Here we use the equalities Dk2 Ek ({0}) = 0 and Ek ({0})Sj = Sj . Similarly, in the case σ m (α) = ∅ for k 6= j using x = Sj Sj∗ x we have Ek ([0, t])Dk2 sm k Ej (δ)x =

1 − qm m S Ej (δ)x. 1−q k m

For k = j, σ m (α) = ∅, and large t we have Ej (δ)Ej ([0, (1−q)t−1+q ]) = Ej (δ) (1−q)qm since δj is bounded, therefore in this case (9) does not depend on t as well as above and obviously converges in Hj as t → ∞. (j) Finally, for a bounded δ and x ∈ Hu we have the following expressions: Dk2 Sα Ej (δ)x

=

Dk2 Sα Ej (δ)x

=

0, α1 6= k; 1 − qm Sα Ej (δ)x, 1−q α = (k, . . . , k, αm+1 , . . . , αl ), αm+1 6= k; | {z } m

(10)

Dk2 Skm Ej (δ)x

=

Dj2 Sjm Ej (δ)x

=

1 − qm m S Ej (δ)x, k 6= j; 1−q k 1 − qm Sjm Ej (δ)( I + q m Dj2 )Ej (δ)x., 1−q

where in the last formula we used 1 − qm 1 − qm I + q m Dj2 )Ej (δ) = ( I + q m Dj2 )Ej (δ). Ej (δ)( 1−q 1−q From (10) we conclude that Dk , k 6= j, is bounded in Hj with ||Dk2 |Hj || = (1 − q)−1 . 1 + r)p kEj (δ)xk, which means that Dj For k = j we have ||Dj2p Sα Ej (δ)x|| ≤ ( 1−q 2 consists of bounded vectors of Dj . 3. Dj is invariant with respect to Sk , Sk∗ , k = 1, . . . , n. The invariance (j) with respect to Sk is obvious. For z = Sα Ej (δ)x with x ∈ Hu we have Sk∗ z = δkα1 Sσ(α) Ej (δ)x if α 6= ∅. For z = Ej (δ)x we have z = Sj Sj∗ z and Sk z = δkj Sj∗ z = δkj Ej (q −1 (δ − 1))Sj∗ x ∈ Dj . 4. Define Ak , k = 1, . . . , n as a closure of Dk Sk from D. Then D ⊂ D(Ak ), D(A∗k ), k = 1, . . . , n and (2) holds on D. This follows directly.

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5. One can easily see that D consists of bounded vectors for the operators Sj∗ Dj2 Sj and using their commutation, for ∆ as well. Therefore ∆ is essentially selfadjoint on D. (j)

Remark 2. In fact it follows from unitarity of Sj on Hu that Hj coincides with (j) the closure of the span of the family {Sα x, x ∈ Hu , α ∈ Λj }, where Λj = {∅, (α1 , . . . , αk ), 1 ≤ αi ≤ n, αk 6= j, k ∈ N}. (j)

Remark 3. It follows from the considerations above that Sα Hu are invariant with (j) respect to Dk2 and if α 6= ∅, α ∈ Λj , then restriction of Dj2 to Sα Hu is bounded. In fact, for nonempty α ∈ Λj and any k = 1, . . . , n one has Dk2 x =

(11)

1 − q mk (α) x, 1−q

x ∈ Sα Hu(j) ,

where the function mk (α) ∈ Z+ , α 6= ∅, is determined by the condition mk (α) mk (α)

(12)

σk

σ

mk (α)+1 mk (α)+1

(α) = α,

σk

σ

(α) 6= α.

(j)

Recall also that Dk2 x = 0, for x ∈ Hu , k 6= j, and with mk (∅) = 0 the formula (j) (11) becomes true for x ∈ Hu and k 6= j also. 2.2. Irreducible unbounded representations. In this section we will give a classification of irreducible unbounded representations of Onq . We will keep notation from the previous section and consider only well-behaved representations of Onq . Let {A1 , . . . , An } be a family of closed operators on H defining a representation π of Onq . Let Ai = Si Ci be the polar decomposition of Ai and Di = Si Ci Si∗ . Denote by Ej (·) the resolution of identity of Dj and let B(R) be the σ-algebra of all Borel subsets of R. Definition 3. The representation π will be called irreducible if the only operator C ∈ B(H) which commutes with all Si , Si∗ and Ei (δ), δ ∈ B(R), i = 1, . . . , n, is a multiple of unity, or equivalently, the space H can not be decomposed into a direct sum of two non-trivial subspaces which are invariant with respect to Si , Si∗ , and Ei (δ), δ ∈ B(R), i = 1, . . . , n. It follows from Lemma 1 that for irreducible unbounded π the representation (j) space H coincides with the closed span Hj of {Sα Hu , α ∈ Λj } for some j ∈ {1, . . . , n}. Lemma 2. Let π be an irreducible unbounded representation of Onq on Hj for some fixed j. Then the restriction Aj |H(j) determines an irreducible representation of O1q (j)

u

on Hu .

Proof. Assume that Huj = Hu1 ⊕ Hu2 where Hui , i = 1, 2, are invariant with respect to Sj , Sj∗ , and Ej (δ) for any Borel δ ∈ R. Let Hi , i = 1, 2, be the closed linear span of {Sα Hui , α ∈ Λj }. Then following arguments in the proof of Theorem 4 we obtain that H = H1 ⊕ H2 where each of Hi is invariant with respect to Sk , Sk∗ , Ek (δ) for any k = 1, . . . , n and δ ∈ R ∈ B(R) contradicting the irreducibility of π.


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1 is fixed. Theorem 5. Let j ∈ {1, . . . , n} and x ∈ (1 + qx0 , x0 ], where x0 > 1−q s Let H be a Hilbert space with orthonormal basis {eα , s ∈ Z, α ∈ Λj } and let Ak , k = 1, . . . , n be linear operators on H given by s 1 − q mk (σk (α)) s s Ak eα = eσk (α) , k 6= j 1−q s 1 − q mj (σj (α)) s Aj esα = eσj (α) , α 6= ∅ 1−q r 1 − qs (13) + q s x es+1 Aj es∅ = ∅ 1−q

where mk (α) are defined by (12). Then {A1 , . . . , Ak } defines an irreducible representation π(x,j) of Onq . Moreover any unbounded irreducible representation is unitarily equivalent to exactly one representation π(x,j) . Proof. Let U be the closure of span of {es∅ , s ∈ Z}. Then U is invariant with respect to Aj , A∗j and the restrictions of these operators to U determine a well-behaved irreducible representation of O1q . Consider the polar decompositions Ak = Sk Ck , and put Dk = Sk Ck Sk∗ , k = 1, . . . , n. Then Dk2 esα

=

Dk2 es∅

=

1 − q mk (α) s eα , α ∈ Λj , α 6= ∅ 1−q Ak A∗k es∅ = 0, k 6= j, s ∈ Z

Ak A∗k esα =

s ∗ s esσj (α) if either k 6= j or α 6= ∅, and Sj es∅ = es+1 ∅ ; Sk eα = δki1 eσ(α) if = e∅s−1 and Sk∗ es∅ = 0, k 6= j. In particular, esα = Sα es∅ , for any nonΛj and s ∈ Z. It is a routine to verify that the conditions of Theorem

Sk esα = ∅, Sj∗ es∅

and α 6= empty α ∈ 4 are satisfied and formulas (13) determine a well-behaved representation π(x,j) of (j) Onq with U = Hu and π(x,j) (ai ) = Ai . Each representation π(x,j) is irreducible. In fact, let C ∈ B(H) be a selfadjoint operator commuting with Sk , Ek (δ), k = 1, . . . , n, δ ∈ B(R). One can easily see n (j) (j) n that Hu = Ej (∆x ), where ∆x = { 1−q 1−q +q x, n ∈ Z}, giving that Hu is invariant q with respect to C. Put C∅ = C|H(j) ; since the representation of O1 defined by the u

(j)

restriction of π(x,j) (aj ) to Hu is irreducible, one has C∅ = λ∅ 1, λ∅ ∈ C. Further, using the commutation of C with all Sk and induction on length of α ∈ Λj , we get (j) that Sα Hu is invariant with respect to C for any α ∈ Λj . Denoting by Cα the corresponding restriction we obtain again by induction Cα = λ∅ 1 for any α ∈ Λj . (j) As H = ⊕α∈Λj sα Hu we conclude that C = λ∅ 1H and π(x,j) is irreducible. Next we show that π(x,j) are non-equivalent representations. Since π(x,j) (ak ), are bounded if j 6= k, we have that π(x,j) is not unitarily equivalent to π(y,k) when j 6= k. Let x, x′ ∈ (1+qx0 , x0 ], j ∈ {1, . . . , n}. Suppose that π(x,j) ≃ π(x′ ,j) . Denote the corresponding representation spaces by H and F and let V : H → F be a unitary operator giving the equivalence of the representations. Then V gives the equivalence of representations of O1q defined by the actions of π(x,j) (aj ) and π(x′ ,j) (aj ) on H (j)

(j)

and F respectively. Consider the decompositions H = Hu ⊕ (Hu )⊥ and F = (j) (j) Fu ⊕ (Fu )⊥ . Recall that summands in these decompositions are invariant with

12


respect to π(x,j) (aj ), π(x,j) (a∗j ) (π(x′ ,j) (aj ), π(x′ ,j) (a∗j ) respectively), the restriction, (j)

(j)

π1 (aj ), of π(x,j) (aj ) (and π1′ (aj ) of π(x′ ,j) (aj )) to (Hu )⊥ ((Fu )⊥ respectively) is bounded and π2 (aj ) = π(x,j) (aj )|H(j) = πx (a), u

π2′ (aj ) = π(x′ ,j) (aj )|F (j) = πx′ (a) u

are π1′

Since π1 and π2 are disjoint and the same and π2′ one has that V ∗ π(x′ ,j) (aj ) implies V = V2 ⊕ V1 and V2 π2 (aj )V2 = π2′ (aj ), hence by ′

π(x,j) (aj )V ∗ = Proposition 1

we get x = x . Finally, by Lemma 2 any irreducible well-behaved representation π of Onq acting on Hilbert space H corresponds to some fixed j = 1, . . . , n such that the restriction (j) of π(aj ) to Hu determines an irreducible well-behaved representation of O1q . Then (j) (j) decomposing H = Hu ⊕ (Hu )⊥ and taking a unitary operator V of the form (j) V = V1 ⊕1, where V1 is unitary acting on Hu such that V1∗ π(aj )|H(j) V1 = πx (a) for u some x ∈ (1 + qx0 , x0 ), we obtain by Remark 3 that V gives the unitary equivalence of π to π(x,j) . Remark 4. Using the same arguments we can describe irreducible representations of Onq such that some of Sj is not a pure isometry (i.e. its Wold decomposition consists of the unitary part). In this case we have the following two possibilities: either the corresponding representation is unbounded as described in Theorem 5, or it is unitarily equivalent to one determined by the following formula s 1 − q mk (σk (α)) eσk (α) , k 6= j Ak eα = 1−q s 1 − q mj (σj (α)) eσj (α) , α 6= ∅ Aj eα = 1−q r 1 Aj e∅ = exp(2πı φj ) (14) e∅ , φj ∈ [0, 1) 1−q on H with orthonormal basis {e∅ , eα , α ∈ Λj }. Representations corresponding to different j = 1, . . . , n or φj ∈ [0, 1) are non-equivalent. In particular, for q = 0 we get a classification of all irreducible representations of On such that one of the generators is not a pure isometry. Acknowledgements The paper was written when the first and the second authors were visiting Chalmers University of Technology and Gothenburg University, Sweden. Warm hospitality and excellent working atmosphere are greatly acknowledged. The research was partially supported by a grant from the Swedish Royal Academy of Sciences as a part of the program of cooperation with the former Soviet Union. The third author was also supported by the Swedish Research Council. References [1] F.Bagarello, Algebras of unbounded operators and physical applications: a survey, Rev. Math. Phys. 19 (2007), no. 3, 231–271. [2] A. O. Barut, R. Raczka, Theory of group representations and applications. Second edition, World Scientific Publishing Co., Singapore, 1986. xx+717 pp.


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